Draw a straight line segment of length 5 cm and name  MN .

Construct an angle of 60° at M such that MN is an arm 

Construct the perpendicular at point N

Mark the point P where the other side of the 60°angle intersects the above perpendicular line.

Here’s how to construct the figure based on your instructions:

### Steps for the construction:

1. **Draw a line segment MN of 5 cm:**
   - Use a ruler and draw a straight line segment of length 5 cm. Label the two endpoints as \( M \) and \( N \).

2. **Construct an angle of 60° at point M:**
   - Place the compass on point \( M \), and draw an arc of any radius that intersects line segment \( MN \).
   - Without changing the radius of the compass, place the compass on the point where the arc intersects \( MN \), and draw another arc.
   - From \( M \), draw a straight line through the intersection of the arcs. This will give you a 60° angle. Label this new ray as \( MP \).

3. **Construct the perpendicular at point N:**
   - To construct a perpendicular to line segment \( MN \) at point \( N \), place the compass at point \( N \) and draw an arc that crosses \( MN \) at two points (on either side of \( N \)).
   - Without changing the compass width, place the compass at each of the two points where the arc intersects \( MN \) and draw two more arcs that intersect each other above \( N \).
   - Draw a straight line through \( N \) and the intersection of the arcs above \( N \). This is the perpendicular.

4. **Mark the point P:**
   - Extend the ray \( MP \) so that it intersects the perpendicular line you just constructed.
   - Mark the intersection point as \( P \).

Now, you should have:
- A line segment \( MN \) of 5 cm.
- A 60° angle at point \( M \) with \( MN \) as one of its arms.
- A perpendicular line at point \( N \), and the point \( P \) where the other side of the 60° angle intersects the perpendicular.

Would you like further details on any step or how to verify these constructions?

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**Related Questions:**
1. How can we verify that the angle constructed is exactly 60°?
2. What are alternative methods to construct a perpendicular at point \( N \)?
3. How can we check that point \( P \) is correctly placed?
4. What is the geometrical significance of the point \( P \) in this construction?
5. How would you modify the construction if the angle at \( M \) was different, say 45°?

**Tip:** Always use a sharp pencil and accurate instruments (compass and ruler) to make precise constructions.

Draw a straight line segment of length 5 cm and name MN. Construct an angle of 60° at M such that MN is an arm. Construct the perpendicular at point N. Mark the point P where the other side of the 60° angle intersects the above perpendicular line.

Learn how to draw a line segment of 5 cm, construct a 60° angle at one end, and draw a perpendicular line at the other end. This tutorial guides you through the steps to find the intersection point P, using core geometry concepts and theorems. Suitable for students in Grades 6-8.

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